Determine how many solutions exist for the system of equations. ${-12x-2y = -14}$ ${6x+y = -9}$
Solution: Convert both equations to slope-intercept form: ${-12x-2y = -14}$ $-12x{+12x} - 2y = -14{+12x}$ $-2y = -14+12x$ $y = 7-6x$ ${y = -6x+7}$ ${6x+y = -9}$ $6x{-6x} + y = -9{-6x}$ $y = -9-6x$ ${y = -6x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -6x+7}$ ${y = -6x-9}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.